Dynamical Systems in Neuroscience:

96 Two-Dimensional Systemsdefines a constant vertical vector field depicted in Fig. 4.3b. The systemẋ = −x ,ẏ = −ydefines a vector field that points to the origin (0, 0), as in Fig. 4.3c, and the systemẋ = −y , (4.3)ẏ = −x (4.4)defines a saddle vector field, as in Fig. 4.3d. Vector fields provide geometrical informationabout the joint evolution of state variables. For example, the vector field inFig. 4.3d is directed rightward in the lower half-plane and leftward in the upper halfplane.Therefore the variable x(t) increases when y < 0 and decreases otherwise, whichobviously follows from the equation (4.3). Quite often however geometrical analysis ofvector fields can provide information about the behavior of the system that may notbe obvious from the form of the functions f and g.4.1.1 NullclinesThe vector field in Fig. 4.3d is directed rightward (x increases) or leftward (x decreases)in different regions of the phase plane. The set of points where the vectorfield changes its horizontal direction is called x-nullcline, and it is defined by the equationf(x, y) = 0. Indeed, at any such point x neither increases nor decreases becauseẋ = 0. The x-nullcline partitions the phase plane into two regions where x moves inopposite directions. Similarly, the y-nullcline is defined by the equation g(x, y) = 0,and it denotes the set of points where the vector field changes its vertical direction.This nullcline partitions the phase plane into two regions where y either increases ordecreases. The x- and y-nullclines partition the phase plane into 4 different regions:(a) x and y increase, (b) x decreases, y increases, (c) x and y decrease, and (d) xincreases, y decreases, as we illustrate in Fig. 4.4.Each point of intersection of the nullclines is an equilibrium point, since f(x, y) =g(x, y) = 0 and hence ẋ = ẏ = 0. Conversely, every equilibrium of a two-dimensionalsystem is the point of intersection of its nullclines. Because nullclines are so important,we consider two examples in detail below (the reader is urged to solve Ex. 1 at the endof this chapter).Let us determine nullclines of the system (4.3, 4.4) with the vector field shown inFig. 4.3d. From (4.3) it follows that x-nullcline is the horizontal line y = 0, and from(4.4) it follows that y-nullcline is the vertical line x = 0. These nullclines (dashed linesin Fig. 4.3d) partition the phase plane into 4 quadrants, in each of which the vectorfield has a different direction. The intersection of the nullclines is the equilibrium(0, 0). Later in this chapter we will study how to determine stability of equilibria intwo-dimensional systems, though in this particular case one can easily guess that theequilibrium is not stable.